Copies of one journey
Take the vector a = (2, 1), 2 across and 1 up. Make the movement a three times, each copy starting where the one before it finished. Across, that is 2 + 2 + 2 = 6; up, it is 1 + 1 + 1 = 3. So a + a + a = (6, 3), and it is written 3a.
A number on its own, such as 3, has a size but no direction. In work with vectors, a number like this is called a scalar, and multiplying a vector by a scalar is called scalar multiplication.
The scalar multiplies every component: 3 × (2, 1) = (3 × 2, 3 × 1) = (6, 3). In letters, k(p, q) = (kp, kq).
Three copies of a = (2, 1), tip to tail, end at (6, 3). That single movement is 3a.
The same direction, three times the length
The arrow a is the hypotenuse of a right triangle with legs 2 across and 1 up. The arrow 3a is the hypotenuse of a right triangle with legs 6 and 3. The second triangle is the first enlarged by scale factor 3, so its angles are the same, and 3a points in exactly the same direction as a.
The enlargement makes every length 3 times as long, the hypotenuse included. By Pythagoras, a has length , which is 2.24 to 2 decimal places, and 3a has length , which is 6.71.
The scalar can be a fraction. points the same way as a and is half as long.
λ = 3: r = a + λd = (7, 4), λ copies of d laid head to tail from A, and sweeping λ through every value traces the whole line
Slide P to λ = 2 and read r = a + 2d
From the point (1, 1), the gold arrow is , the vector d = (2, 1) multiplied by the scalar , the Greek letter lambda. At it is 3d = (6, 3). Drag its head along the dashed line: below the arrow is shorter than d, at it vanishes, and below 0 it points the other way. Every multiple of d stays on the dashed line, which runs in the direction of d. The sum printed at the top gives the point where the gold arrow ends.
A negative scalar
Multiplying by −1 changes the sign of both components, which reverses the vector: (−1)a = −a = (−2, −1), the negative of a.
Any negative scalar does two things at once. −2a = (−2 × 2, −2 × 1) = (−4, −2): the 2 makes the arrow twice as long, and the minus sign turns it to point the opposite way.
A length is never negative. −2a is long, the same as 2a; the minus sign changes only the direction. So if a vector a is 5 long, then −3a is 3 × 5 = 15 long. In general, the length of ka is the size of k, ignoring its sign, times the length of a.
a = (2, 1) and −2a = (−4, −2): twice the length, pointing the opposite way.
Multiples lie on one line
Draw a, 2a, 3a and −2a from the origin. They end at (2, 1), (4, 2), (6, 3) and (−4, −2), and every one of those points lies on the same straight line through the origin, the line . Points that lie on one straight line are called collinear.
Two vectors are parallel exactly when one is a scalar multiple of the other. To test a pair, find the multiplier for each component and see whether it is the same number. For (4, 2) and (6, 3), the multipliers are 6 ÷ 4 = 1.5 and 3 ÷ 2 = 1.5, so (6, 3) = 1.5 × (4, 2) and the two are parallel. For (2, 3) and (4, 5), they are 4 ÷ 2 = 2 and 5 ÷ 3, which is not 2, so the two are not parallel.
a, 2a, 3a and −2a, all drawn from the origin, lie along the one line .
A scalar times a sum
Multiplying a sum by a scalar gives the same result as multiplying each vector first: k(a + b) = ka + kb. For example, 2((1, 3) + (2, −1)) = 2(3, 2) = (6, 4), and 2(1, 3) + 2(2, −1) = (2, 6) + (4, −2) = (6, 4) as well.
The usual mistakes
Multiplying only one component. 3 × (2, 1) is (6, 3), not (6, 1): changing one component alone turns the arrow off its own line.
Forgetting the reversal. −2 × (2, 1) is (−4, −2), not (4, 2).
Giving a length as negative. If a is 5 long, −3a is 15 long, not −15.
Adding the scalar to the length. Scaling by 3 multiplies the length by 3; it does not add 3.
Halfway, and a third of the way
In the application below, each village is given by its position vector: the vector from the origin, a railway station, to the village. The point halfway between A and B has position vector , half of the sum, so each of its components is the mean of the two coordinates.
A point a third of the way along the road from A to B is reached by going to A and then a third of the vector from A to B. That vector is b − a, so the point has position vector .
Worked example: A Phone Mast Halfway Between Two Villages, and a Second a Third of the Way Along
Question On a map with its origin O at a railway station and distances in kilometers, village A has position vector a = 29 and village B has position vector b = 143. (a) A phone mast M is built halfway between the villages. Find its position vector and its distance from the station. (b) A second mast R is built on the straight road from A to B, one third of the way from A. Find its position vector.
1.The midpoint's position vector is half the sum of the two: m = 12(a + b) = 121612 = 86.
The midpoint has half the sum of the two position vectors: m = 121612 = 86. 2.(a) M has position vector 86, and its distance from the station is √82 + 62 = √100 = 10 km.
(a) M has position vector 86 and is √82 + 62 = 10 km from the station. 3.The vector along the road from A to B is AB = b − a = 143 − 29 = 12−6.
Along the road, AB = b − a = 12−6. 4.One third of the way from A is one third of that vector: 1312−6 = 4−2.
One third of the way from A is 1312−6 = 4−2 on from A. 5.(b) r = a + 13AB = 29 + 4−2 = 67. Check: from R to B is 8−4, twice 4−2, so R is twice as far from B as from A.
(b) r = a + 13AB = 29 + 4−2 = 67.
Answer: (a) m = 86, 10 km from the station; (b) r = 67
Common mistakes
- Writing the midpoint as 12(b − a) = 6−3. That is half the step from A to B; it has to be added to a before it is a position.
- Taking 13b for the second mast. A third of the position vector of B is a third of the way from the station to B, not a third of the way from A to B.